G.B.
Stop rescuing them, allow the to experience the consequences of their habits. I would not pick up anything they mess up, I would also let them spit and then when it is still there the next time they do it they might spill it on themselves.
If you continue to clean up after them they will continue to treat you as the maid their dad sleeps with.
I would also consider finding my own place. If you have limited income or are a low income family please get on the waiting list now for low income housing to get out of there.
Just because you move out does not mean you can't still be with this man, you may just have to wait out his kids. Then let the man know if he wants you that having a cleaning crew come in to sanitize the house would be a nice housewarming present.
I have a dear friend who married the man of her dreams. He does a brisk business of collecting junk cars and selling the parts off of them from his back yard.... She knew going in that she would be the main breadwinner for the family and would be continuing to work full time. She could not get along with the new hubbies young son. She was yelled at, belittled, etc...the hubby backed her up completely. She finally decided to get her own apartment for a while to let the son have some peace. Once her grew up a bit and could accept her she was able to move home and they lived much more happily. He moved out on his own a year or so later and considers her his closest thing to a mother. Her moving out for a while and continuing to date the boyfriend allowed the son to accept her more slowly and gave him time to get to know her in a different light. She became much less threatening to his relationship with his dad and he was able to start treating her with respect and ended up loving her.